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Assessing the prognostic ability of the stratified Cox proportional hazards model

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... City Heart Study: A Tybj rg-Hansen, ECAT Angina Pectoris Study: F Haverkate, ... Tuomainen; Malm : B Hedblad, P Lind; MONICA/KORA-Augsburg: H Loewel, W Koenig, ... – PowerPoint PPT presentation

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Title: Assessing the prognostic ability of the stratified Cox proportional hazards model


1
Assessing the prognostic ability of the
stratified Cox proportional hazards model
  • Lisa Pennells
  • Ian White
  • Angela Wood
  • Stephen Kaptoge
  • Nadeem Sarwar
  • The Fibrinogen Studies Collaboration

2
Outline
  • Background
  • The Fibrinogen Studies Collaboration (FSC) The
    data
  • Modelling
  • Measures to estimate prognostic ability
  • Measures we have chosen to quantify prognostic
    value
  • Adaptation of measures for use with a stratified
    model
  • Adaptation of measures to display their value
    over time
  • Conclusions

3
The Fibrinogen Studies Collaboration (FSC)
  • Aim to amalgamate available information on the
    role of Fibrinogen in Coronary Heart Disease
    (CHD) and other Cardiovascular endpoints
  • Raised usual plasma fibrinogen levels thought to
    increase risk of cardiovascular disease.
  • Combination of data from 31 long-term prospective
    studies
  • 154 211 adults without a history of previous
    cardiovascular disease
  • Median follow up 8.42 years (total 1.38
    million person-years)
  • Several dozen characteristics reported at
    baseline including
  • Established cardiovascular risk factors e.g.
    Cholesterol, blood pressure
  • Main endpoint on which analysis focused First
    CHD event (n7118)

Fibrinogen Studies Collaboration, JAMA 2005
294(14)1799-1809.
4
Modelling
  • Stratified Cox regression
  • where
  • the probability of surviving beyond time t
    given covariates x and strata value k
  • the baseline survival function for strata
    k
  • In analysis of FSC data we used a Cox model that
    was stratified by cohort, sex and in the case of
    one study, trial arm

5
Hazard ratios obtained from the stratified model
6
Assessing the prognostic power of Variables
  • The need for measures of Prognostic ability
  • Hazard ratios dont tell us how useful a model
    is for making predictions
  • Statistical significance of hazard ratios depends
    largely on sample size
  • Measure needed - Interpretable
  • - comparable across data sets of different
    sizes
  • Previous methods used in cardiovascular
    epidemiology
  • Most commonly used Area Under ROC Curve (AUROC)
  • Only appropriate for binary data!
  • Challenge for FSC
  • Identify methods appropriate for use with
    survival data
  • Adapt for use with the stratified model

7
Measures Chosen
  • Explained variation
  • Proportion of variation in the outcome explained
    by the covariates in the model.
  • Range 0 to 1 (values lt 0.4 common for survival
    data)
  • Schemper and Hendersons V
  • Designed to be consistent under with random
    censoring
  • Measures of discrimination
  • Summarise a models ability to discriminate
    between levels of subject risk (or risk ranking)
  • Harrells C-index
  • Equivalent to the AUROC (already well know in CV
    epidemiology)
  • Royston and Sauerbreis D
  • Several favourable properties (including
    simplicity)

Schemper and Henderson, Biometrics 2000 56(1).
Harrell FE, Jr. et al. JAMA 1982 247(18).
Royston and Sauerbrei. Stat.Med. 2004 23(5).
8
V Original
  • difference between observed survival and that
    predicted using a model without covariates,
    averaged a) over all subjects at each failure
    point
  • b) over all failure points (with weights)
  • as for but using a model with covariates
  • Weights when averaging across failure points to
    give and

Schemper M, Henderson R. Biometrics 200056(1)
9
V adapted for the stratified model
  • Inaccuracies for each individual are determined
    according to strata specific predicted survival
    probabilities
  • Inaccuracies are averaged over time using strata
    specific weights
  • V over time
  • At each time point of interest (e.g. at 5 year
    intervals) only inaccuracies calculated at
    failure points occurring before this time are
    averaged (using relevant weights)

10
Problem with V for the stratified model
  • Problem
  • V increases initially with time. Hence, if
    combining information from studies which differ
    in follow up time, inclusion of information from
    studies at time points after their study period
    has ended leads to downward bias in the value of
    V
  • Solution
  • At each time point, only include information
    from studies that are still running

11
A well behaved V over time
12
V from 4 studies with different follow up
13
Combining the 4 studies regardless of follow-up
14
Effect of combining V from studies of different
follow up
a) Combining regardless of follow-up
b) Using only information from current studies
0.07
15
D Original
  • Motivation
  • To quantify the observed spread of disease risk
    across the range of estimated
  • Calculation
  • Fit CPH model
  • Transform to give standard normal order
    rank statistics (rankits - formed using Bloms
    approximation)
  • Multiply rankits by a factor of to
    give zi (i1n subjects)
  • Fit a CPH model to these values D is the
    coefficient of z from this second model
  • Interpretation
  • log hazard ratio comparing two equal-sized
    prognostic groups based on dichotomising
  • a continuous prognostic index ( )
  • D adapted for the stratified model
  • Replace steps 1 and 4 above by the fitting of a
    stratified model

Royston P, Sauerbrei W. Stat.Med. 2004 23(5)
16
C-index Original
  • Definition The probability that, for a randomly
    selected pair of subjects, the person who
    fails first has the worse prognosis.
  • Range 0.5 (discrimination no better than chance
    prediction) to 1(perfect discrimination).
  • Estimation Class all pairs in which the subject
    with the shorter participation time fails as
  • Concordant, agreeing in rank of and order
    of failure
  • Discordant, opposite in rank of and order of
    failure
  • Undecided, tied in either category
  • The numbers of class 1, 2 and 3 are then counted
    to give , and respectively, and
  • combined in the calculation of the C-index

Harrell FE, Jr. et al. JAMA 1982 247(18).
17
C-index for the Stratified model
  • A stratified model is fitted to the data
  • Comparison of pairs is restricted to occur within
    strata
  • Numbers of pairs are summed overall
  • D and the C-index over time
  • A model is fitted to the whole data
  • is extracted (and in the case of D
    transformed)
  • The data set is then censored at each time point
    before the comparison of pairs (in the case
    of C) and the fitting of the second model (in the
    case of D)

18
Final versions of V, D and the C-index over time
for FSC data
19
Conclusions
  • For survival data it is important to use a
    measure of prognostic value which adequately
    handles its characteristics.
  • Each of V, D and the C-index appeared promising
    in principle, for adaptation to the stratified
    model.
  • We found the measure of explained variation V to
    be difficult to apply when data are combined from
    several studies with differing durations of
    participant follow-up. Confidence intervals also
    increased throughout follow-up for our data.
  • The two other measures considered, D and the
    C-index, were more applicable in such
    circumstances.

20
Acknowledgments
  • Professor John Danesh (University of Cambridge)
  • Members of the Fibrinogen Studies Collaboration
  • AMIS JB Kostis, AC Wilson Atherosclerosis Risk
    in Communities Study AR Folsom, K Wu, L
    Chambless BIP Registry M Benderly, U Goldbourt
    Bruneck Study J Willeit, S Kiechl Caerphilly
    Study JWG Yarnell, PM Sweetnam (this prospective
    cohort study was undertaken by the former UK
    Medical Research Council Epidemiology Unit (South
    Wales) and was funded by the MRC its data
    archive is maintained by the Department of Social
    Medicine, University of Bristol) Cardiovascular
    Health Study M Cushman BM Psaty, RP Tracy (see
    http//chs-nhlbi.org for acknowledgements)
    Copenhagen City Heart Study A Tybjærg-Hansen,
    ECAT Angina Pectoris Study F Haverkate, MPM de
    Maat, SG Thompson Edinburgh Artery Study
    Edinburgh Claudication Study FGR Fowkes, AJ Lee,
    FB Smith FINRISK 1992 Hemostasis Study V
    Salomaa, K Harald, V Rasi, E Vahtera, P
    Jousilahti, J Pekkanen Framingham Study R
    D'Agostino, WB Kannel, PWF Wilson, G Tofler, D
    Levy GISSI Prevenzione R Marchioli, F
    Valagussa Göteborg 1913 Göteborg 1933 A
    Rosengren, L Wilhelmsen, G Lappas, H Eriksson
    GRIPS Study P Cremer, D Nagel Honolulu Heart
    Program JD Curb, B Rodriguez, K Yano Kuopio IHD
    Study JT Salonen, K Nyyssönen, T-P Tuomainen
    Malmö B Hedblad, P Lind MONICA/KORA-Augsburg H
    Loewel, W Koenig, HW Hense Northwick Park Heart
    Study I TW Meade, JA Cooper, B De Stavola, C
    Knottenbelt Northwick Park Heart Study II GJ
    Miller, JA Cooper, KA Bauer, RD Rosenberg Osaka
    Study S Sato, A Kitamura, Y Naito, H Iso
    Platelet Activation and Inflammation Study V
    Salomaa, K Harald, V Rasi, E Vahtera, P
    Jousilahti, T Palosuo PRIME Study P
    Ducimetiere, P Amouyel, D Arveiler, AE Evans, J
    Ferrieres, I Juhan-Vague, A Bingham PROCAM
    Study H Schulte, G Assmann Quebec
    Cardiovascular Study B Cantin, B Lamarche, J-P
    Després, GR Dagenais Scottish Heart Health
    Study H Tunstall-Pedoe, GDO Lowe, M Woodward
    Speedwell Y Ben-Shlomo, G Davey Smith Strong
    Heart Study V Palmieri, JL Yeh Thrombosis
    Prevention Trial TW Meade, P Brennan, A
    Rudnicka, C Knottenbelt, JA Cooper US Physicians
    Health Study P Ridker VITA F Rodeghiero, A
    Tosetto West of Scotland Coronary Prevention
    Study J Shepherd, GDO Lowe, I Ford, M Robertson
    Whitehall II E Brunner, M Shipley Zutphen
    Elderly Study EJM Feskens, D Kromhout.

21
Composition of V
,
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